The propagation of elastic waves in strongly heterogeneous media using subgrid modeling approach is studied. The local elastic parameters and the mass density have essential variations in some interval of scales at each spatial point. To approximate a strongly heterogeneous medium, we have started from the modified Kolmogorov theory in terms of the ratios of smoothed fields. The correlated fields of the elastic stiffness and of the mass density have been represented mathematically by the Kolmogorov multiplicative cascades. The wave propagates over a distance that is of the same order as the typical wavelength of a source. The 2D averaged elastic equations are obtained if the wavelength is large as compared with a maximum scale of the medium heterogeneities. If a medium is assumed to satisfy the improved Kolmogorov similarity hypothesis, the expression for the effective elastic parameters is especially simple. It has been shown that small-scale heterogeneities affect the wave propagation.